WITH A LITTLE HELP FROM
OBJECTIVES
At the end of this lesson, you will be able to:
1. Apply trigonometry to solve real world problems
and everyday situations
2. Describe processes and situations accurately, as
well as justify and link arguments coherently.
3. Read, understand and interpret data from a variety
of sources in order to solve different tasks
4. Identify the theorem and the trigonometric ratio to
be used in each situation
5. Use the calculator correctly
6. Collect, organize and display data to answer
questions
7. Graphically represent each situation in a precise
and accurate way
CONTENTS
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ac
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ad
1. Measure of angles in radians.
2. Trigonometric ratios of an angle
3. Use of the calculator to find the trigonometric
ratios and angles
4. Use of formulae and trigonometric ratios to solve
triangle problems
5. Use of trigonometry to calculate heights.
6. Use of formuale and trigonometric ratios to solve
everyday situations.
7. Use of instruments to measure angles
ASSESSMENT CRITERIA
side
Sections:
1. What’s Trigonometry good for?
2. Solving Surveying Problems
3. Final Task: Tutorial for partners
1 Use trigonometric ratios of angles greater than 90º
correctly.
2. Use instruments of measure to calculate heights
3. Use the calculator correctly in order to solve
trigonometric ratios and calculate angles, which has
the same trigonometric ratio
4. Define and demonstrate trigonometry theorems
5. Solve problems using the appropriate ratio
6. Describe the process followed to solve different
situations, accurately and using the specific
terminology
7. Graphically represent each situation
MATEMÁTICAS
- ADAPTACIÓN ALUMNADO ALTAS CAPACIDADES -
1
SECTION 1: WHAT’S TRIGONOMETRY GOOD FOR ?
1. 1
Read the following text and find out what trigonometry is good for.
Click on the link below
http://www-istp.gsfc.nasa.gov/stargaze/Strig1.htm
1. 2. Read again and answer the following questions:
- What is a basic example of a trigonometry problem?
- Why are triangles used as the basis for trigonometry calculations?
- What trigonometry projects are mentioned? How were calculations made? What for?
3. Try to define a similar surveying project where trigonometry could be used to solve a problem. (First, describe the situation or problem, then explain how to solve it)
2
With a little help of trigonometry
SECTION 2: SOLVING SURVEYING PROBLEMS
Trigonometry, as you know, is used for surveying and engineering, in order to measure land surfaces, horizontal and vertical distances between two points, heights of mountains and river widths.
Now, we will learn to use formulas to solve two of these common situations.
1.
Calculate the width of a river which cannot be crossed due to geographical issues or problems.
In this situation, we can mark points
A, B, C on one of the river banks,
bearing in mind that the distance
between A and B is 50 metres, as
it is shown in figure 1, and the angle CAB must be 90 º. On the line
AC, we visualize a point D, which is
on the other side of the river. With
the help of a theodolite, located
on point B, we measure the angles
ABC and CBD. Then calculate the
width of the river.
Figure 1
2.
Calculate the height of a
mountain.
To measure the height of a mountain,
we stand on the points A and B, 275
meters apart from each other, and
using a theodolite we take measures
which are shown in the figure 2.
Once you have done this, calculate
the height of the mountain.
Figure 2
Now, think of a similar situation. Make a sketch and describe it to your partners:
MATEMÁTICAS
- ADAPTACIÓN ALUMNADO ALTAS CAPACIDADES -
3
FINAL TASK: TUTORIAL FOR PARTNERS
Although these problems can be solved using trigonometric ratios of right triangles, it is easier and
faster to use the results of trigonometry for angles in general. Now, study the following sections and
carry out our final task: to prepare a tutorial for your partners explaining to them how to solve a
problem integrating the new concepts and theorems and giving them examples of everyday situations where they can be applied.
Process:
1.
Read and study the sections below.
2.
Do the activities suggested for each of them
3.
After that, prepare a slideshow or a presentation using prezi, openoffice or powerpoint, to
make trigonometry easier for your partners. It has to include introduction of trigonometry with
definitions, explanations and pictures, as well as:
a. Definition of radian and its measure in degrees, minutes and seconds
b. Number of radians that a circle has and conversion of radians into degrees, and viceversa.
(First, show to them using a model and then explain the demonstration)
c. Use of theodolites
d. Definition and demonstration of sine theorem
e. Choose or invent and then solve a problem using the sine theorem, which can help your
partners understand this theorem better.
Remember: You can use pictures, diagrams, graphs or drawings to enhance the presentation.
4.
Prepare a script for your presentation and a quiz to test your partners understanding
Requirements:
1. Duration of presentation: 10 minutes
2. The slideshow must include all the sections listed above
3. You will hand in the activities and exercises carried out, as well as
the script for your presentation and the presentation itself.
Some useful phrases
a. Welcome the audience
b.Introduce yourself and the topic
c.Outline the structure of your preIntroduction
sentation
d. Give instructions about
questions
Body
The body should be well structu- Let’s start …
red, divided up logically, with plen- Now let’s move on to …
ty of carefully spaced visuals.
I’d like to illustrate this by showing you …
I’d like to turn to something completely different
This picture/diagram shows…As you can see,.
a. Sum up
b. Thank your audience
Conclusion c. Invite questions
4
My presentation is about ...
The main points I will be talking are ...
First, I’d like to talk about …
Secondly, …
In conclusion, ...
Now, to sum up...
So let me summarize/recap what I've said.
That’s the end of my presentation. Thank you for
your listening. Have you got any questions?
With a little help of trigonometry
Degree and Radian measurement of angles
We know that once around a circle is 360º (a circle has three hundred and sixty degrees),
1o = 60´( one degree has sixty minutes) and 1´= 60´´( one minute has sixty seconds).
A radian is a unit of angle widely used in maths and science. Using radians to measure
angles seems unnatural at first. However, when
angles are stated in radians the constant pi tends
to disappear from the equations, and this greatly
simplifies calculation.
A radian is the angle of an arc created by
wrapping the radius of a circle around its
circumference.
Now, we need to know the relationship
between both definitions, in order to convert
degrees to radians, and viceversa. We will solve
this process by answering these two questions:
a)
How many radians are there in a
circle?
b)
What formula can give us the
relationship between radians and degrees?
Follow the instructions and build the material necessary to explain to your partners
the correspondence between degrees and radians.
a.1 First, draw a circumference with a radius of 10 cm.
a.2 Secondly, put nails around the circumference contour and then stick it in a cork
board or a noticeboard.
a.3 After that, using a flexible tape measure (for example, the one used for sewing) and
holding onto the nails, draw a 10 cm long arc on the circumference. The angle obtained
measures 1 radian (you must get a figure similar to the one shown above)
a.4 Continue marking all the radians which you can draw on that circumference.
a.5 Measure the arc remaining to complete the 360 º of the circumference.
a.6 Finally, relate the number of degrees which the circumference has and the
(decimal) number of radians which you have obtained.
a.7 In conclusion, the formula to convert degrees to radians, and viceversa, is:
Converting radians into degrees and viceversa
π
The circumference of a circle is given by 2 r so the radius, r, fits around the circumference of a circle exactly 2 times and there are 2
radians in a complete circle.
π
π
2 radians= 360º
π
π
radian= 180º
Converting radians into degrees and vice versa:
In order to convert radians into degrees we make use of the fact that
radians equal
one half circle, or 180º.
This means that if we divide radians by , the answer is the number of half circles.
Multiplying this by 180º it will tell us the answer in degrees.
180º
The formula to convert radians into degrees is: degrees= radians --------
π
π
π
MATEMÁTICAS
- ADAPTACIÓN ALUMNADO ALTAS CAPACIDADES -
5
7RFRQYHUWGHJUHHVLQWRUDGLDQVwe will isolate radians in this formula and we will
ʌ
get radians= degrees . ----------
180º
(Adapted from: http://www.teacherschoice.com.au/Maths_Library/Angles/Angles.htm)
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2
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a) 35º 30´30´´
b) 46º45´
3.
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c) 125º 25´52´´
Positive and negative angles
Angles are positive angles if the
terminal ray rotates counterclockwise around the vertex from the initial ray.
6
Angles are negative angles if the
terminal ray rotates clockwise around
the vertex from the initial ray
With a little help of trigonometry
Goniometric circumference.
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LV DW WKH RULJLQ of coordinates and the UDGLXV
LVRQH
Any point on the circumference is at distance
1 from the origin, therefore, if we represent an
acute angle with the vertex at the origin of
coordinates and one side on the positive X axis
OX, the value of VLQH coincides with the RUGLQDWH of
the point of intersection of the other side with the
goniometric circumference. In the same way,
the value of FRVLQHcoincides with the abscise
of this point and the tangent with the value of
the RUGLQDWH GLYLGHG E\ WKH DEVFLVH We can
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y
sin a= y cos a=1 tan a= ------x
Where a is an angle between 0º and 360º.
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Consider a circle which has the radiusU .
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CURRÍCULO INTEGRADO DE LAS LENGUAS
7
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Depending on the quadrant the signs of sine, cosine and tangent of the angle a are
different.
sine
cosine
tangent
First quadrant
+
+
+
Second quadrant
+
-
-
Third quadrant
-
-
+
Fourth quadrant
-
+
-
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Sine Theorem.
If A, B and C denote the angles of a triangle, and a, b, and c the lengths of the sides
opposite them, the following result holds:
b
c
a
---------- = ---------------- = ---------------sin C
sin A
sin B
Note::KHQ\RXXVHWKHVLQHUXOHWR¿QGDPLVVLQJDQJOH$DOZD\VFKHFNZKHWKHU
(180° - A)is a possible solution as well.
(it is because both angles, A and (180º-A), verify that sinA=sin(180º-A) )
8
With a little help of trigonometry
Cosine Theorem
When we know the lengths of the three sides of a triangle but none of the angles, we
cannot use the sine formula to find the angle because there will be two unknown angles whichever pair of fractions we chose. In these cases we need a different formula.
It is called the cosine rule.
If A, B and C denote the angles of a triangle, and a, b, and c the lengths of the sides
opposite them, the following result holds:
?
Note that, in each version of cosine rule, the side on the left and the angle on the right
have the same letter.
sine
Choosing which rule to use:
cosine
Use the sine rule when:
a) you know two sides and one angle (not between the two sides) and want a
2nd angle (3rd angle is now obvious!).
b) you know two angles and one side and want a 2nd side.
Use the cosine rule when:
a) you know three sides and want any angle.
b) you know two sides and the angle between them and want the 3rd side.
Practice makes perfect I: Sine and Cosine Theorems
1.
Read the sections about Sine and Cosine Theorems
2.
Search for information on the internet or textbooks and study the demonstrations of these theorems. You can find them at:
a)
http://www2.warwick.ac.uk/services/elearning/mathsfit/trigonometry/3/
b)
Mathematical Textbooks (for example, Matemáticas I, for 1 bachillerato)
MATEMÁTICAS
- ADAPTACIÓN ALUMNADO ALTAS CAPACIDADES -
9
Crucial points in order to solve problems
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Make sure that you check whether any missing angles can be found using geometry
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whether (180° – A) is also a solution.
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Make sure you check that all the units are the same so you are not mixing, say, kilometres and metres.
4. &KHFN\RXUFDOFXODWRUPRGH
Make sure your calculator is in degrees mode and not in radians mode, or viceversa.
5. Be careful not to lose accuracy through rounding.
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Practice makes perfect II
1.
Click on the interactive sine/cosine rule and do the exercises in section 11.3
(Bearing, the examples solved and the suggested activities, from 1 to 10, too)
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REVISION
Now, let’s check understanding:
Click on KWWSZZZZDUZLFNDFXNVHUYLFHVHOHDUQLQJPDWKV¿WWULJRQRPHWU\DVWT
pdf and do the 08/7,3/(&+2,&(7(67
10
With a little help of trigonometry
Theodolites
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QRWHVDQGWKHQFRPSOHWHWKHIROORZLQJFXHFDUG
Picture:
Description:
Use or function:
Fields and professions where it is used:
Brief history:
Remember: you will use this information to prepare the tutorial for your partners.
CURRÍCULO INTEGRADO DE LAS LENGUAS
11
A FEW TIPS FOR YOUR PRESENTATION
Step 1 Planning and Preparation
Prepare! Prepare! Prepare!,
because preparation is everything.
Step 2 Structure
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Introduction
Body
Conclusion
Step 3 Visual aids
(photos, images, tables, etc)
“A picture is worth 1,000 words”
Golden rule: use one image to give one message
Step 4 Signposting
When you read a book, you know where you are (the chapter, the page, the section,..)
but when you give a presentation, your audience does not know where they are- UNLESS
you tell them
Here are just a few useful expressions
My presentation is about …
First, I’d like to talk about ..
Second, …
This picture/slide/photo
shows
That’s the end of my presentation. Thank you for
listening. Have you got any
questions
Step 5 Delivery
Talk to the audience, don’t read and
make sure your audience can hear you clearly.
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use a loud and clear voice
speak naturally
make eye contact
vary pitch and tone
use appropriate words and grammar
A
few
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words more use
fu
and p
Let’s begin by
hrase l
s
First, …
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Now, we’ll move on to
Next,…
Let’s look now at..
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For example
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In conclusion,
Let’s summarise
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First, … second
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12
With a little help of trigonometry
ASSESSMENT CRITERIA
Criteria
Properly
organized to
complete project
Managed time
wisely
Acquired needed
knowledge
Process
Below Avg.
Satisfactory
Excellent
(1,2,3, 4)
(5, 6, 7)
(8,9,10)
(5, 6, 7)
(8,9,10)
(1,2,3, 4)
(1,2,3, 4)
(5, 6, 7)
Criteria
(8,9,10)
Product
Below Avg.
Satisfactory
Excellent
(1,2,3, 4)
(5, 6, 7)
(8,9,10)
Requirements of
the project.
(1,2,3, 4)
(5, 6, 7)
Organization and
structure
(1,2,3, 4)
(5, 6, 7)
(8,9,10)
(1,2,3, 4)
(5, 6, 7)
(8,9,10)
(1,2,3, 4)
(5, 6, 7)
Explanation.
Mechanics of writing and speaking.
Maths mechanics
Use of visuals.
Creativity.
Demonstrated
knowledge
Points
(8,9,10)
(8,9,10)
(8,9,10)
(1,2,3, 4)
Points
(5,6, 7)
.
TOTAL:
MATEMÁTICAS
- ADAPTACIÓN ALUMNADO ALTAS CAPACIDADES -
13